Trigonometry Concepts with Interactive Diagrams

In trigonometry, we start with a right triangle, which always has one 90-degree angle.Every right triangle has three important sides: the hypotenuse, which is always the longest side and opposite to the right angle, the opposite side, which is across from our angle theta, and the adjacent side, which is next to our angle theta.Let's measure these sides. For this example, our adjacent and opposite sides are both 3 units, making the hypotenuse 3 root 2 units by the Pythagorean theorem.From these sides, we can define our three main trigonometric ratios. Let's start with sine.Next is cosine, which is adjacent over hypotenuse.Finally, tangent is opposite over adjacent.A key property of these trigonometric ratios is that they remain constant for similar triangles of different sizes.Whether our triangle is small or large, as long as the angles remain the same, the ratios of the sides will be identical.To help remember these ratios, we use the mnemonic device SOH-CAH-TOA.SOH reminds us that Sine equals Opposite over Hypotenuse.CAH tells us that Cosine equals Adjacent over Hypotenuse.And TOA helps us remember that Tangent equals Opposite over Adjacent.The unit circle is a circle with radius 1 centered at the origin.As we move a point around this circle, its coordinates give us the sine and cosine values for each angle.Watch how the x-coordinate represents cosine and the y-coordinate represents sine as we move around the circle.Let's look at some special angles and their values.Here's a table of common angle values and their sine and cosine ratios.Notice how the values repeat every three hundred and sixty degrees, or two pi radians, showing the periodic nature of these functions.